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acoustic reflection imaging do not have the function of azimuthal measurement due to a symmetric source, so they can not be used to evaluate the azimuthal character of borehole formation. In this paper, a 3D fi nite difference method was used to simulate the acoustic fi elds in a fl uid-fi lled borehole generated by a traditional monopole source and a phased arc array. Acoustic waveforms were presented for both cases. The analysis of the simulated waveforms showed that different from the monopole source, the acoustic energy generated by the phased arc array transmitter mainly radiated to the borehole in a narrow azimuthal range, which was the key technique to implement azimuthal acoustic well logging. Similar to the monopole source, the waveforms generated by the phased arc array in the fluid-filled borehole also contain compressional (P) waves and shear (S) waves refracted along the borehole, which is the theoretical foundation of phased arc array acoustic well logging. Phased arc array, acoustic fi eld, acoustic well logging, azimuth, fi nite difference method Key words: develop a special downhole source (Qiao et al, 2006a), which 1 Introduction can radiate acoustic energy azimuthally onto the sidewall. With the development of complicated hydrocarbon Azimuthal radiation may fundamentally resolve the diffi cult exploration, azimuthal acoustic logging has been urgently problems of azimuthal measurement ability, evaluation of needed (Li and Chu, 1997). The acoustic monopole source, heterogeneous and anisotropic formations, and improvement widely used in acoustic well logging, can be treated of the signal-to-noise ratio. Qiao et al (2008) designed approximately as a point source, so the acoustic field a phased arc array acoustic transmitter and obtained its generated by the point source in a fluid-filled borehole is radiation directivity curve by both experimental measurement axially symmetric. In a fluid-filled borehole surrounded by and numerical calculation. The results showed that the a soft formation, shear waves can not be generated by the acoustic energy generated by the phased arc array transmitter monopole source. To implement shear wave acoustic logging was in a specifi c azimuthal range. This study is to calculate in a soft formation, excited flexural motion was presented the acoustic fi eld generated by a phased arc array transmitter (Brie et al, 1998). The subsequent acoustic well logging in a fl uid-fi lled borehole and to compare it with the acoustic technology combined quadrupole and dipole sources with fi eld generated by a traditional monopole source. the monopole source, which expanded the functions and application scope of acoustic well logging (Kessler and 2 Working principle of a phased arc array Varsamis, 2001; Tang and Cheng, 2004; Tang, 2004). Unlike transmitter the monopole source, the dipole and quadrupole sources are directional, but they have limited azimuthal measurement The schematic diagram of a 14-element acoustic phased ability. For example, the directivity curve of the dipole source arc array transmitter is shown in Fig. 1. Each element is a is like a “∞” shape. Another problem is multiple solutions piezoelectric vibrator. If the five elements centering on the in formation inversion. A great deal of research on acoustic No. 4 element are excited, the acoustic travel time from the well logging with phased linear arrays has been carried No. 2 element and the No. 3 element to line AB are τ and τ 2 1 out (Che et al, 2005; 2006; Chen and Xu, 2008; Qiao et al, respectively. 2002; 2003). It is shown that the signal-to-noise ratio of the When the phased arc array is activated, the No. 2 and the measurement signal can be improved by using a phased linear No. 6 elements are excited simultaneously and this moment is array, but the acoustic field generated by the phased linear marked as zero time. Then the No. 3 and the No. 5 elements array is also axially symmetric. The key to making acoustic are excited simultaneously at τ −τ , and the No. 4 element 2 1 well logging able to achieve azimuthal measurements is to is excited at τ . In this way, the wave fronts generated by all active elements can reach line AB at τ simultaneously. It seems like a 5-element equivalent linear phased array *Corresponding author. email: aclab@cup.edu.cn centering on the No. 4 element, with each element located Received December 30, 2008 226 Pet.Sci.(2009)6:225-229 w T w v w v § · w v w v yy y y x z ¨ ¸ O 2P g (5) yy ¨ ¸ 14 1 2 w t w x w y w z w y © ¹ w v § · w T w v w v w v zz x z z ¨ ¸ O 2P g (6) zz ¨ ¸ w t w x w y w z w z © ¹ 11 4 w T w v § w v · xy y 10 5 x ¨ ¸ P (7) 1 ¨ ¸ w t w y w x © ¹ 9 6 8 7 τ w T § w v w v · xz x z (8) B ¨ ¸ w t w z w x © ¹ Fig. 1 Schematic diagram of a phased arc array acoustic transmitter w T § w v · w v yz y (9) ¨ ¸ along a line parallel to the y-axis. If at a specifi c moment the ¨ ¸ w t w z w y © ¹ No. 2–No. 6 elements emit, at the next moment the No. 3–No. 7 elements emit, and in the same way different elements Assuming the unit size in the staggered grids is Δx×Δy×Δz are excited in turn at different moments, then azimuthally (Fig. 2 shows 1/8 of a unit in the staggered grids), the normal scanning radiation of acoustic energy can be achieved. stress T (m=x, y, z) is in the center of the unit, and the mm time step is Δt. Then in the staggered grids, the wave field 3 Numerical calculation components in the discrete space and time domains are defi ned as follows: 3.1 Method The finite difference method is extensively used to simulate acoustic fields in acoustic well logging (Liu et ªº 11 1 §· § · § · al, 1996; Lin et al, 2006; He et al, 2006). In this paper, the Ti (,j,k) T i ' x,j' y , k' ' z;n t mm mm¨¸ ¨ ¸ ¨ ¸ «» 22 2 ¬¼ ©¹ © ¹ © ¹ acoustic field generated by a phased arc array in a fluid- filled borehole is simulated by the 3D finite difference (10) ,, mx y,z mx ,,y z method. Because the elastic wave equations of fluid can be regarded as a special case of those of a solid, the following fi nite difference equations are derived from the elastic wave ª 1 º § · (11) equations of solids. When referring to fluid, the elastic T (i, j , k ) T' i x,' j y,¨ k '¸ z; n' t xy xy« » © ¹ ¬ ¼ coeffi cient μ=0. In Cartesian coordinates, for a homogeneous elastic medium with mass density of ρ=ρ(x, y, z) and Lame constants ª º of λ=λ(x, y, z) and μ=μ(x, y, z), v , v , and v are particle § 1· x y z n (12) T (i, j, k ) T ' i x, j ' y,' k z; n' t ¨ ¸ xz xz« » velocity components in the x, y, and z directions, T , T , and xx yy 2 © ¹ ¬ ¼ T are normal stresses of x, y, and z axes, T , T , and T are zz xy xz yz shear stresses in the planes of xy, xz, and yz, and g (j = x, y, jj ª 1 º § · z) is the acoustic source density of stress rate (Pa/s). The fi rst- n (13) T (i, j, k ) T ¨ i' ¸ x' , j y,' k z; n' t yz yz « » order partial differential equations (Liu et al, 1996; Lin et al, © ¹ ¬ ¼ 2006) for the velocity and stress are given below: w T w v w T w T xy ª 1 1 1 º x xx xz § · § · § · n 1 / 2 (1) (14) v (i, j, k ) v ' i x,¨ j '¸ y,¨ k '¸ z;¨ n ¸ ' t x x« » w t w x w y w z 2 2 2 © ¹ © ¹ © ¹ ¬ ¼ w v w T w T w T y xy yy yz (2) w t w x w y w z ª 1 1 1 º § · § · § · n 1 / 2 (15) v (i, j, k ) v ¨ i ' ¸ x,' j y,¨ k '¸ z;¨ n ¸ ' t y y« » w T w v w T w T 2 2 2 yz z xz zz © ¹ © ¹ © ¹ ¬ ¼ U (3) w t w x w y w z § w v · w T w v w v w v ª º xx x z x § 1· § 1· § 1· (4) n 1 / 2 O ¨ ¸ 2P g (16) xx v (i, j, k ) v i ' x, j ' y,' k z; n ' t ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ z z« » w t w x w y w z w x © ¹ 2 2 2 © ¹ © ¹ © ¹ ¬ ¼ θ Pet.Sci.(2009)6:225-229 227 yz Sidewall v T 0° z xz Δz/2 xy (a) Δy/2 Δx/2 mm v (m=x, y, z) Fig. 2 Diagram of 1/8 of a unit in the staggered grids Sidewall After discretization (second-order in the time domain and eighth-order in the space domain), Eq. (1) is approximated as below: nn 1/ 2 1/ 2 v i,,jk v i, jk, xx ' t nn ªº aT (, i l j,k) T (i l 1, j,k) ¦ lxx xx ¬¼ U ' x l 0 ' t nn aTªº (,i j l 1,k) T (,ij l,k) lxy xy ¬¼ U ' y l 0 ' t nn aTªº (,i j,k l 1) T (,i j,k l) ¦ lxz xz ¬¼ U ' z l 0 (17) In Eq. (17), the weight coefficients of a ( l=0-3) are (b) 1225/1024, -245/3072, 49/5120, and -5/7168, respectively. Similar approximation can be applied to Eqs. (2)–(9). 3.2 Model well The dimensions of the model well used for numerical Sidewall calculation is 1.2 m×1.2 m×1.5 m. The distance from the borehole axis to the outer wall of the model well is 0.6 m, and the borehole diameter is 0.2 m. The P-wave velocity and S-wave velocity of the borehole formation are 4500 m/s and 2500 m/s respectively, and the density of the borehole formation is 2200 kg/m . The velocity and density of the borehole fl uid are 1500 m/s and 1000 kg/m respectively. The space step is Δx = Δy = Δz = 5 mm, the center frequency of pulse source f = 12 kHz, and the time step is Δt = 2.5e-7 s. Fig. 3(a) illustrates the locations of the acoustic transmitter and receiver array in the xy-plane of the fluid- fi lled borehole. “T” is the acoustic transmitter with a radius of 35 mm, “R” is the azimuthal receiver array centering on the acoustic transmitter with a radius of 90 mm. The azimuthal angle between the two adjacent receivers is 3.75°. The center of the phased arc array is in the direction of θ = 0°. Fig. 3(b) and Fig. 3(c) show the locations of the receiver array corresponding to the monopole source and the phased arc (c) array transmitter in the fluid-filled borehole respectively, in Fig. 3 Locations of acoustic transmitters and receiver array in the fluid- which the axial distance of two adjacent receivers is 250 mm. fi lled borehole 228 Pet.Sci.(2009)6:225-229 Pet.Sci.(2009)6:225-229 229 Che X H, Zhang H L, Qiao W X, et al. Numerical study on scanning 4 Conclusions radiation acoustic field in formations generated from a borehole. 1) Different from the traditional monopole source, Science in China, Ser. G Physics, Mechanics & Astronomy. 2005. 48(2): 247-256 a phased arc array acoustic transmitter is able to radiate He F J, Tao G and Wang X L. Finite difference modeling of the acoustic energy to the borehole in a narrow azimuthal acoustic field by sidewall logging devices. Chinese Journal of range, so evaluating the azimuthal character of the borehole Geophysics. 2006. 49(3): 923-928 (in Chinese) formation is feasible. Kes sler C and Varsamis G L. A new generation crossed dipole logging 2) Similar to the symmetric source, the waves generated tool: design and case histories. SPE Annual Technical Conference by the phased arc array in the fluid-filled borehole also and Exhibition. New Orleans, Louisiana. 30 September–3 October, contain refracted P waves and S waves of the formation, 2001. SPE 71740 and this is the theoretical foundation of the phased arc array Lin W J, Wang X M and Zhang H L. Acoustic wave propagation in a acoustic logging. borehole penetrating an inclined layered formation. Chinese Journal 3) Phased arc array azimuthal acoustic logging may of Geophysics. 2006. 49(1): 284-294 (in Chinese) become a new generation of acoustic logging technology, Liu Q H, Schoen E, Daube F, et al. A three-dimensional fi nite difference simulation of sonic logging. J. Acoust. Soc. Am. 1996. 100(1): 72-79 but the complexity of transducers and the corresponding Li Z B and Chu Z H. Present situation and tendency of research on well electronic circuit must be faced. logging in China. Chinese Journal of Geophysics. 1997. 40(Supp.l): 333-343 (in Chinese) Acknowledgements Qia o W X, Chen X L, Du G S, et al. Laboratory simulation on acoustic This work is supported by the National Natural Science well-logging with phased array transmitter. Chinese Journal of Acoustics. 2003. 22(4): 329-338 Foundation of China (Grant Nos. 10534040, 40574049 and Qia o W X, Che X H, Ju X D, et al. Acoustic logging phased arc array 40874097), the Research Fund for the Doctoral Program and its radiation directivity. Chinese Journal of Geophysics. 2008. of Higher Education (Grant No. 20070425028) and the 51(3): 939-946 (in Chinese) Foundation of State Key Laboratory of Petroleum Resource Qia o W X, Du G S and Chen X L. Feasibility of application of linear and Prospecting, China University of Petroleum (Grant No. phased array acoustic transmitters to acoustic well-logging. Chinese PRPDX2008-08). Journal of Geophysics. 2002. 45(5): 714-722 (in Chinese) Qia o W X, Ju X D, Chen X L, et al. Downhole acoustic arc array with References controlled azimuthal directivity. China Patent. ZL 03137596.0. 2006a (in Chinese) Bri e A, Endo T, Hoyle D, et al. New directions in sonic logging. Oilfi eld Qia o W X, Ju X D, Che X H, et al. Multipole acoustic logging Review. 1998. 40-55 transmitter structurized by annular array. Journal of China University Che n X L and Xu X K. Application of linear phased-array technique of Petroleum (Edition of Natural Science). 2006b. 30(5): 33-36 (in to estimation of acoustic properties of formation in acoustic Chinese) logging of cased hole. Acta Petrolei Sinica. 2008. 29(5): 777-781 Tan g X M and Cheng A. Quantitative Borehole Acoustic Methods. (in Chinese) Netherlands: Elsevier Publishing Company. 2004. 1-27 Che X H, Qiao W X, Ju X D, et al. The acoustic field in fluid-filled Tan g X M. Imaging near-borehole structure using directional acoustic- borehole generated by linear phased array transmitter with 2D wave measurement. Geophysics. 2004. 69(6): 1378-1386 spectrum technique. Progress in Natural Science. 2006. 16(10): (Edited by Hao Jie) 1282-1287 (in Chinese)
Petroleum Science – Springer Journals
Published: Jul 23, 2009
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